Can You Ski Off The Top Of The Train In Polar Express?

Skiing off of Polar Express

I love the emails I get from readers of this blog. Yesterday morning, I got the following from Shane Schieffer of FitTrip and I just saw it as I was grinding through my morning backlog from the weekend.

“Hi Brad, Happy holidays. My friend and I were watching Polar Express with our kids last night, and he commented on the impossibility of skiing down the top of a train towards the engine, even on a crazy steep slope. I know you like to geek out on software and things such as figuring out the algorithms behind slip numbers, how about physics? The question was bothering me so I took a shot at it this morning. Thought I’d share the result. I bet users of your forum could top my work, or find errors, or extend it to further interesting observations. Kind of a fun holiday geeky thing to contemplate, so I thought I’d send it your way.”

This reminded me of a great story from Amy. On her very first trip to Dallas from Boston to meet my parents, she was sitting on a plane next to a guy who was doing a bunch of math sheet of paper. She asked him what he was doing. He said he was calculating how much fuel the airplane needed to get from Boston to Dallas. It turned out to be a guy named Christopher Couch, who was an undergrad at MIT that had crossed paths with me for some reason I can no longer remember. She and Chris had a great time on the plane together talking about all kinds of nerdy things. The entire memory made smile.

Oh – here’s the answer.

“Assuming just the train engine (not cars, and cargo) that the polar express was modeled after, which weighs 361,136kg and has an approximate cross sectional surface area of 10m^2, at freezing (0-deg Celsius) and 1atm of pressure (sea level) on a -128.5 gradient (what the sign said in the movie which equates to a -52 degree slope) assuming frictionless tracks, and a hill tall enough to induce free fall, where no braking nor engine acceleration is applied, and the coefficient of drag is based on a rectangular shape, the train would be traveling 1,434 mph. This is clearly much faster than the terminal velocity of a 6-foot tall man with a boy on his shoulders (say a combined 3m), standing up, weighing a combined 240lbs, with a cross sectional surface area of 2.25m^2, whose terminal velocity would only be 52.5 mph on that same slope. Of course the train itself would create a wind draft that would lessen the difference, but either way the man and boy are going off of the back end of that train. Unless the man is a spirit…”

If you aren’t familiar, the guy who writes the xkcd comic has a really cool Tuesday feature called What If? He takes an oddball physics question, not unlike the one answered here, and tries to figure out the impact. His first was what is the effect of a baseball thrown at the speed of sound. A few weeks ago he discussed whether you could achieve flight with machine guns as a propellant. He’s a former NASA engineer and the site is here: http://what-if.xkcd.com/

so best case you need a vertical ramp half a kilometer high. (without wind resistance) (say a third of a mile)

With wind resistance the ramp needs to be infinite (in principal – terminal velocity is an asymptote you never reach)

But this ski slope will slow the train down more than the person (you can ignore wind resistance -= why because at these speed the skier is inside the drag envelope of the train)

Now assume terminal velocity of train on the flat (with a ski-scraper sized ski-ramp on top big enough ski-ramp) << 20 mph

So we have three dimensional extremes predicated on the power of the train (if on the flat) and the size of the ski-ramp and the slope of the track.

As the slope gets steeper the length of the ramp gets shorter (the train is pulling the ramp downwards – making it longer from the skiers perspective)

But as the slope gets steeper the train goes faster (this does not stop the skier accelerating because they are probably shrouded in a boundary layer of the train (air moving with the train and the ski ramp).

Limit condition – vertical free-fall train – do you fall relative to train – yes if the train is carrying a finite static pressure wave with it that draws you down in its envelope – and it will if you stick a ski-ramp on it taller than the person.

Between these two positives it is certainly possible (a minima with a positive second differential lies between the two limits – regardless of engine power ) so for any mix of parameters a possible zone can be found.

By the way – the ski-ramp would want to be pretty well enforced but note the mass of the re-inforcement has no effect – it accelerates under gravity just like anything else.

So bottom line – YES and without further ado – check this video – its totally cooler than that

Make sure to match like with like. The skier in a speed skiing suit would fall faster than 156 mph off of a cliff, due to the lack of drag from his skis, provided he was able to keep the speed profile during the fall.

Brad, I don’t remember enough about the movie to answer this.. Why did you choose to represent the train in essentially a freefall, rather than using a speed record (113 MPH for a steam engine. Now the question is of drag from the skis. Is the drag from the skis enough to allow the train’s speed to counteract wind resistance on the adult+child, but not enough to keep gravity from making the duo accelerate faster than the train and move toward the engine.

Jon – must concede your freefall consideration – But I think you will find windage losses from railway cabooses much more significant than ski friction.

Fun exercise though if only to train (pun intended) the mind on fresh approaches to *Not VERY Important* Questions. Makes approaches to real problems so much more innovative (a nd reduces the fear of being out on a limb)

– so Good Call Brad

Shane Schieffer

You’ve asked a great question. I made an absurd assumption. I made the decision quickly under the knowledge that a human is near free-fall after 10 seconds; and the clip in the film lasts about that long. Of course I obviously didn’t anticipate the incredible speed that a train would reach at terminal velocity. Once I got the answer I didn’t mind the absurdity of that particular assumption since it yielded an interesting observation and made it clear that air resistance becomes the force that determines the result. I now view the question similar to whether a crouton set atop an apple would fall faster than the apple. This feels more obvious. The answer is that all things fall at g (or the cosine of 52 * g in the train example) until drag takes over which would happen very quickly with the skier and very slowly with the train. When said that way it kind of feels like an apple hitting me on the head…

A train at 1,434 mph is doing Mach 2 or about 600 metres per second so Navier-Stokes equations no longer apply (big time) but again for fun. Train accelerates at best at a steady 1g (quite unlikely) taking over 60 seconds. Distance travelled vertically = 18 kilometers
Considerably higher than twice the height of Mt Everest.

The cost of building the tracks will be high but wind ressistance will be lower up there in the Stratosphere. There will be no problem covering the train in ice !

Shane Schieffer

Nice. Calculating the height of a mountain that could induce terminal velocity is an interesting extension of thought. I used that assumption for simplicity but didn’t realize just how unrealistic the scenario becomes. Good work.